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PaperclipPerfector  5d ago (text post)   26493 thread views

Culture War Roundup for the week of February 17, 2025

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There's one particular question that I ask LLMs, courtesy of my maths PhD cousin: "Is the one-point compactification of a Hausdorff space itself Hausdorff?" The correct answer is yes, or so I'm told.

Are you just asking it as a yes/no question? This is a standard question that a first-year undergrad could be asked to check that they understood the definitions, and it's unlikely that the answer wouldn't be in the training set somewhere. For example, I quickly fed it to a Q4_K_M quantised Qwen2.5-3B (that's on the level that you could run on a smartphone nowadays), and it completed

Q: Is the one-point compactification of a Hausdorff space itself Hausdorff?

A:

with

Yes, the one-point compactification of a Hausdorff space is itself Hausdorff.<|endoftext|>

edit: See @wlxd's discussion for why the correct answer is actually "No". In fact, Qwen2.5-3B is almost perfectly on the edge: the log-odds of the first token of an alternative answer that goes "No, the one-point compactification of a Hausdorff space is not necessarily Hausdorff.<|endoftext|>" is only about 0.21 lower, so the probability of it being chosen is about e^-0.21 or 0.81 times the probability that it would pick the "Yes...". (Think 45% No to 55% Yes.)

I've replied to him. I think there's been error on my part, my cousin's explanation probably included an elaboration on the additional qualifications of the answer.

Most of the reasoning models seem to be saying yes (I didn't keep track of precise numbers), and if they said no, then @wlxd 's response is one they acknowledge as correct.

The point is that this answer is just incorrect. There are non Hausdorff one point compactifications of Hausdorff spaces. You need additional assumption of local compactness for it to be true.

You are right, but I don't think that was "the point", given that @self_made_human apparently was led to believe that it is yes (and seemed to treat that answer being given as a success criterion).

(I was actually in the process of writing up another response as I had realised it is not true, after I fed the question to DeepSeek-R1's Qwen7B distill to reason through and found that it choked as it tried to conjure up compact neighbourhoods that I didn't see the grounds for existing, but I hadn't gotten to the point of having a good counterexample yet)

It's likely PEBKAK on my part. My cousin had explained the reasoning almost a year back (eons in LLM time) and I'd likely forgotten the finer qualifications and only remembered the yes or no bit.

I give a counter example in my other comment.

Ah, thanks, that works.

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